2.1 Modified target plane

Let us suppose the nominal orbit with initial conditions X^* has a close approach with some planet, which is defined as a passage within a given distance d. Practical limits for the selection of the distance d are that it must not be too large to result in the possibility of simultaneous close approach with two planets, but large enough to allow the detection of close approaches for the nearby orbits; values between 0.03 and 0.1 AU are suitable for the terrestrial planets. The distance from the planet r(t,X^*) has at least one minimum lower than d, at some time t=t^*. A practical algorithm to compute t^* is the regula falsi applied to search for the zeros of the function dr/dt; it is easy to discriminate the minima from the maxima of r by checking the sign of dr/dt. If there are multiple minima, a particular one has to be selected or ddecreased until only one minimum remains.

We define as Modified Target Plane (MTP) the plane through the center of the approached planet and perpendicular to the velocity vector of the encountering body at the closest approach time t^*. This definition is the same used by [Yeomans and Chodas 1994] (they call it target plane), and is a modification of the target plane used in Öpik's theory of close encounters [Öpik 1976]; the latter is the plane through the planet and perpendicular to the unperturbed velocity vector. These two planes are of course close to each other for shallow encounters. Once an MTP has been selected, we can use cartesian orthogonal planetocentric coordinates $(\xi,\eta,\zeta)$ such that at closest approach on the nominal orbit

$$\xi(t^*,X^*)=0 \ \ \ ;\ \ \{\displaystyle d \over \displaysty... ...\ \{\displaystyle d \over \displaystyle dt}\zeta(t^*,X^*)=0\ ;$$

this is possible because at the nominal orbit the MTP crossing is orthogonal; we shall further assume that the crossing of the MTP is transversal, that is with non zero velocity:

$${\displaystyle d \over \displaystyle dt}\xi(t^*,X^*)\ne 0\ ,$$

that is, the $\xi$ axis is along the planetocentric velocity vector.

The choice of the orthogonal coordinates $(\eta, \zeta)$ in the MTP is arbitrary; for the figures of this paper, we have selected the axes in such a way that $\zeta(t^*,X^*)=0\;,\; \eta(t^*,X^*)>0$, that is, $\eta$ is along the vector from the planet to the nominal closest approach position. If the MTP is crossed only once during the close approach, we can select an orientation such that $d\xi/dt>0$. Low velocity encounters, which can result in multiple minima of the planetocentric distance and/or multiple crossings of any selected MTP, would introduce complications in the geometry of the confidence boundaries which are beyond the scope of this paper, focused on encounters of NEO with the terrestrial planets; in the cases we are interested in, $d\xi/dt$ is larger than the planetocentric parabolic velocity. For orbits close to, but different from, the nominal one, the time at which the MTP is crossed can be different from t^*, and is a function of the initial conditions: t=t(X), implicitly defined by the equation

$$\xi(t(X),X)=0 \ \ \ ;\ \ \t(X^*)=t^* \ .$$

By the ordinary implicit function theorem, the gradient of the function t(X) can be computed by

$${\displaystyle \partial t \over \displaystyle \partial X} = -... ...{\displaystyle \partial \xi \over \displaystyle \partial X}\ .$$

The map onto the MTP is expressed by the functions $\eta(X), \zeta(X)$, that can be shown to be differentiable as follows, provided the MTP crossing is transversal:

$$\left .{\displaystyle \partial \eta \over \displaystyle \part... ...playstyle \partial t \over \displaystyle \partial X} (t^*,X^*)$$

and by using the above formula for the gradient of t(X), and the property that the nominal orbit crosses the MTP orthogonally:

$$\left .{\displaystyle \partial \eta \over \displaystyle \part... ...tyle \partial \eta \over \displaystyle \partial X}(t^*,X^*)\ .$$

In the same way:

$$\left .{\displaystyle \partial \zeta \over \displaystyle \par... ...yle \partial \zeta \over \displaystyle \partial X}(t^*,X^*)\ .$$

In practice this means that, given the state transition matrix, solution of the variational equation along the nominal orbit at time t=t^*, the partial derivatives of the map on the MTP are easily computed. As for the computation of the map itself for an orbit with initial conditions $X\neq X^*$, the solution t(X) of the equation $\xi(t(X),X)=0$ is obtained by iterating a regula falsi as above.

We shall therefore assume from now on that the position of the intercept on the MTP is a 2-dimensional vector $Y=[\eta,\zeta]^T$, and that Y is a function of the initial conditions X, known together with its partial derivatives:

$$Y=F(X)\ \ \ ;\ \ \\ Delta Y= Y-Y^*= DF(X^*)\, \Delta X + \ldots$$

where $\Delta X= X-X^*$, the Jacobian matrix DF is a $2\times 6$ matrix and the dots stand for the higher order terms.